I’m sorry, I don’t know what it means to install on the same database or a different one.
Both sites are installed on the same hosting account. I guess that means they’re in different sub folders, but I don’t know.
This is what I did:
I logged into my hosting account, bought the second domain, downloaded WP, logged into this second domain’s WP admin panel, published my first blog, and then when I logged out of it and tried to login to the first domain’s WP admin panel, it redirected me to the second one.